Jacobian Math Homework: Chain Rule and Determinants of Composition

Then, use the rule det(AB) = det(A)det(B) to show that the resulting matrix is the product of the two Jacobians.In summary, the problem asks to use the chain rule and the rule det(AB) = det(A)det(B) to show that the Jacobian of the composition T2 o T1 is the product of the Jacobians of T1 and T2. This involves computing the derivative of T_2 \circ T_1 and using the rule det(AB) = det(A)det(B) to show the desired result.
  • #1
phrygian
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Homework Statement



Suppose that P, Q, and R are regions in R2, and suppose T1 : P -> Q and T2 : Q -> R are
dierentiable. Use the (multivariable) Chain Rule and det(AB) = det(A)det(B) to show that the Jacobian of the
composition T2 o T1 is the product of the Jacobians of T1 and T2.


Homework Equations





The Attempt at a Solution



So if I say T1 is given by x = g(u,v) and y = h(u,v) and T2 is given by u = i(s,t) and v = j(s,t) , I know how to find the two jacobians T1 and T2 but am confused how do you find the jacobian of the composite transformation T1 o T2?
 
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  • #2


Follow their instructions: use the chain rule to compute the derivative of [tex]T_2 \circ T_1[/tex].
 

1. What is the Jacobian matrix in math?

The Jacobian matrix is a mathematical tool used to describe the relationship between two sets of variables. It consists of the first-order partial derivatives of one set of variables with respect to another set of variables. In other words, it represents the slope of the tangent line at a specific point in a multivariate function.

2. How is the Jacobian matrix used in physics?

In physics, the Jacobian matrix is used to calculate the change in one set of variables with respect to another set of variables. It is commonly used in the study of fluid dynamics, quantum mechanics, and electromagnetism, among other areas.

3. What is the difference between a Jacobian matrix and a Hessian matrix?

The Jacobian matrix is a first-order derivative matrix that describes the relationship between two sets of variables, while the Hessian matrix is a second-order derivative matrix that describes the curvature of a multivariate function. In other words, the Jacobian matrix is a linear approximation of a function, while the Hessian matrix provides information about the second derivatives of a function.

4. How do I calculate the determinant of a Jacobian matrix?

The determinant of a Jacobian matrix can be calculated by taking the partial derivatives of the function with respect to each variable, arranging them in a matrix, and then taking the determinant of that matrix. It is an important calculation in the study of multivariate functions and is used to determine the change in volume, area, or other quantities when the variables change.

5. Can the Jacobian matrix be used in optimization problems?

Yes, the Jacobian matrix can be used in optimization problems to find the maximum or minimum of a multivariate function. It is used in the optimization process to determine the direction and magnitude of the gradient of the function, which is a crucial step in finding the optimal solution.

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